On the mailnews application main toolbar we have GetMsg button. You can a notice
a small dropmarker icon on the righthand side lower corner of that button. If
you click on that you will notice 'Get All New Messages'. This will fetch new
messages for all those accounts which have been authenticated (i.e., logged in
OR have the password remebered by Password Manager) in that session.
Other ways to trigger this action :
1. MailNews App|File|Get New Messages For|Get All New Messages
2. While the focus is on MailNews app, do Ctrl+Shift+t